1. INTRODUCTION

Light manipulation at deep subwavelength scales has attracted considerable interest in recent years because of its applications in diverse areas of nanophotonics, ranging from optical sensing [1–6] to light harvesting [7,8], photodetection [9–12], and nonlinear optics [13–19]. Surface polaritons are pivotal in this effort, as they can exhibit in-plane wavelengths that are substantially smaller than the wavelength of light propagating at the same frequency [20–23]. This concept has been pushed to the atomic-scale limit with the emergence of two-dimensional (2D) materials capable of sustaining different types of surface polaritons [20], including graphene plasmons [21,22,24], atomically thin metal plasmons [23,25–27], and optical phonons in hexagonal boron nitride (hBN) [28], which have proved to be promising for applications in optical sensing [29,30], infrared spectrometry [12,31], and nonlinear nanophotonics [19,32].

Achieving good coupling between light and polaritons is critically important for practical applications involving external light sources. Advances in this direction have relied on the use of optical cavities, such as nanoscale tips, which have become a popular choice to achieve large near-field enhancements and increased coupling to surface modes for imaging and spectroscopy purposes [33–35]. However, this approach still lies far from the ideal situation of complete light-to-polariton coupling at designated spatial locations. In this context, it has recently been shown that coupling through a near-the-surface dipolar scatterer results in a light-to-polariton scattering cross section ${\sim}\lambda _{\rm p}^3/{\lambda _0}$, expressed in terms of the polariton and light wavelengths, ${\lambda _{\rm p}}$ and ${\lambda _0}$, respectively [36]. The coupling cross section is consequently small compared with the area of a diffraction-limited light spot ${\sim}\lambda _0^2$, thus resulting in a poor photon-to-polariton conversion ratio ${\sim}\lambda _{\rm p}^3/\lambda _0^3$.

When a lossless dipolar scatterer is placed in a homogeneous dielectric environment, it can produce complete extinction of a focused light beam [37]. A practical realization of this idea led to the demonstration of substantial extinction by a single two-level molecule, which was required to be held under cryogenic conditions to prevent inelastic losses [38], followed by a recent demonstration of coupling between a molecule and a plasmonic nanoparticle [39]. Now, by bringing the scatterer close to a surface in order to assist coupling to surface modes, it is precisely this coupling that unavoidably introduces losses from the point of view of the scatterer, an effect that leads in turn to a reduction in the scattering strength and the associated coupling efficiency [36]. We now ask the question whether the efficiency of light-to-polariton coupling assisted by a small scatterer can be increased by separating it from the surface, so that it is positioned at an optimum finite distance defined by the compromise between an efficient interaction with the evanescent fields associated with surface modes and the resulting reduction in surface-related inelastic losses that quench the strength of the scatterer. This idea resembles the so-called critical coupling conditions, whereby complete absorption of a light plane wave by a structured planar surface is produced when radiative and nonradiative loss rates are made equal [40–42]. We expect that the scatterer-surface distance provides a knob to vary the balance between radiative and nonradiative loss processes, but the overall efficiency of the proposed scheme remains uncertain.

 

Fig. 1. Optimum light coupling to surface polaritons mediated by a small particle. (a) Illustration of the configuration under consideration. A small particle, described through its induced dipole $\textbf{p}$, is placed at a distance $a$ from the planar surface of a homogeneous film. A light beam with optimized angular profile couples maximally to polaritons supported by the film. (b) Angular phase and amplitude profile of the optimum light beam needed to achieve complete coupling to a semi-infinite silver surface with axially symmetric p-polarized light, which renders $\textbf{p}\parallel \hat{\boldsymbol z}$. We take a particle-surface separation $a = 200\;{\rm nm} $ and a light wavelength ${\lambda _0} = 417\;{\rm nm} $. (c) Electric field amplitude (color plots) and Poynting vector field lines (white arrows) of the incident focused beam in the absence (left) and presence (right) of the surface + scatterer system.

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In this work, we demonstrate that a small scatterer placed at an optimum distance in front of a planar surface can produce complete coupling from a suitably shaped focused light beam into surface modes. We based our results on rigorous analytical theory for small dipolar scatterers, and further corroborate their validity through numerical simulations for finite-size silicon particles, which act as nearly perfect scatterers at their dipolar Mie resonances and can produce ${\gt}80\%$ absorption of a focused light beam by a silver surface. We offer detailed general prescriptions on the required angular profile of the incident light beam and the characteristics of the scatterer, which depend on the composition and optical properties of the surface material. We illustrate this concept with examples of optimum coupling to planar dielectric waveguide modes, as well as phonons in hBN and plasmons in graphene and ultrathin metallic films. This study opens a new route toward complete coupling between light and surface polaritons based on realistic structures incorporating an engineered structure placed in front of the surface. Additionally, our results can be readily applied to two-level quantum emitters, such as optically trapped atoms and molecules in the vicinity of a polariton-supporting surface, which can equally assist both complete optical coupling and strong polariton-mediated interaction among emitters placed at designated positions.

The type of system under consideration is sketched in Fig. 1a, consisting of a scatterer situated at a distance $a$ from a homogeneous planar surface. Throughout this work, we take small scatterers that have axial symmetry with respect to the out-of-plane direction (i.e., the combined particle-substrate geometry also has axial symmetry). Figure 1b shows an example of the angular profile of the incident beam that is required to achieve complete absorption by a semi-infinite silver surface (see below for details of the calculation), whereas Fig. 1c portrays the incident (left) and total (right) electric near-field amplitude. We superimpose on the latter the associated Poynting vector field lines (white arrows), which are clearly redirected along the surface in the presence of the particle, indicating that absorption of the incident light is dominated by coupling to surface plasmons. In practice, a spatial light modulator coupled to a high-numerical-aperture objective could be used to project the complex angular profiles of far-field amplitude and phase that are needed by the incident beam for optimum coupling.

2. EXTREMAL SCATTERING PROPERTIES OF A SMALL PARTICLE

Before introducing a planar surface, we consider a particle in a homogeneous environment and concentrate on externally supplied monochromatic light of frequency $\omega$, writing the optical electric field at a position $\textbf{r}$ and time $t$ as $\textbf{E}(\textbf{r},t) = 2{\rm Re}\{{\textbf{E}(\textbf{r}){{\rm e}^{- {\rm i}\omega t}}} \}$. The field amplitude in a constant $z$ plane fully contained inside a homogeneous medium of permittivity ${\epsilon _{\rm h}}$ can be rigorously decomposed into plane waves as

A. Maximum Focal Field, Scattering, and Absorption by a Point Particle in a Homogeneous Medium

We first explore the light profile needed to maximize the intensity of the electric field component along a direction determined by the complex unit vector $\hat{\boldsymbol n}$ (normalized as $\hat{\boldsymbol n} \cdot {\hat{\boldsymbol n}^*} = 1$) at a focal point $\textbf{r} = 0$ within the $z = 0$ plane for fixed total incident power ${{\cal P}^{{\rm inc}}} = {{\cal P}^ +} + {{\cal P}^ -}$. By imposing the vanishing of the functional derivatives $\delta \{|{\hat{\boldsymbol n}^*} \cdot \textbf{E}{(0)|^2}/{{\cal P}^{{\rm inc}}}\} /\delta \beta _{{\textbf{k}_\parallel}\sigma}^{\nu *} = 0$, we find $\hat{\boldsymbol n} \cdot {\hat {\textbf{e}}}_{{\textbf{k}_\parallel}\sigma}^\nu \;{{\cal P}^{{\rm inc}}} - \hat{\boldsymbol n} \cdot {\textbf{E}^*}(0) (c/2\pi k) {k^\prime _z}\beta _{{\textbf{k}_\parallel}\sigma}^\nu = 0$ (see Appendix A for details of a similar calculation), which leads to the optimum solution

Analogous results are obtained for the maximum focal magnetic field $|{\hat{\boldsymbol n}^*} \cdot \textbf{H}(0)|$. Combining Eq. (1) with Ampère’s law, $\textbf{H} = \nabla \times \textbf{E}/{\rm i}k$, we find

Incidentally, following this approach, we find that the ratio between the electromagnetic energy density at the focus, ${\cal U} = [{{\epsilon _{\rm h}} |\textbf{E}{{(0)|}^2} + |\textbf{H}{{(0)|}^2}}]/8\pi$, and the incident power is automatically maximized for any linear combination of the external light coefficients given by Eqs. (3) and (6), leading to

B. Limits to Absorption and Scattering

We now consider a small particle placed at $\textbf{r} = 0$, whose optical response is dominated by an electric polarizability ${\alpha _{\rm E}}$ along a direction $\hat{\boldsymbol n}$. In response to external illumination, a dipole

Interestingly, these coefficients have the same form as Eq. (3), so for a suitably chosen polarizability ${\alpha _{\rm E}}$, scattering by the particle can completely suppress transmission of the incident light, which is therefore fully reflected. This happens, for instance, for a beam propagating toward positive $z$’s with $\beta _{{\textbf{k}_\parallel}\sigma}^ + = - \beta _{{\textbf{k}_\parallel}\sigma ,{\rm dip}}^ +$ and $\beta _{{\textbf{k}_\parallel}\sigma}^ - = 0$; indeed, by inserting these coefficients into Eq. (1), and this in turn into Eq. (8), applying the identity $\sum\nolimits_\sigma \int_{{k_\parallel} \lt k^\prime} ({{\rm d}^2}{\textbf{k}_\parallel}/{k^\prime _z})(\hat{\boldsymbol n} \cdot {\hat {\textbf{e}}}_{{\textbf{k}_\parallel}\sigma}^ +)({\hat{\boldsymbol n}^*} \cdot {\hat {\textbf{e}}}_{{\textbf{k}_\parallel}\sigma}^ +) = 4\pi k^\prime /3$, we find the condition ${-}\alpha _{\rm E}^{- 1} = 2{\rm i}{k^{\prime 3}}/3{\epsilon _{\rm h}}$ for perfect beam reflection. Now, we note that the optical theorem imposes the constraint ${\rm Im}\{- \alpha _{\rm E}^{- 1}\} \ge 2{k^{\prime 3}}/3{\epsilon _{\rm h}}$ [45], with the equality implying a lossless scatterer (i.e., when radiative losses account for the full value of ${\rm Im}\{- \alpha _{\rm E}^{- 1}\}$); in addition to this, our scatterer must be resonant (i.e., ${\rm Re}\{- \alpha _{\rm E}^{- 1}\} = 0$). We conclude that a focused beam with the angular profile of a dipolar far-field [Eq. (3)] is completely reflected by a lossless resonant dipolar scatterer, as previously predicted using arguments based on numerical simulations [37].

The condition for complete absorption (i.e., full cancellation of the incident field by the induced dipole field) is given by $\beta _{{\textbf{k}_\parallel}\sigma}^ \pm = - \beta _{{\textbf{k}_\parallel}\sigma ,{\rm dip}}^ \pm$, which—upon insertion again into Eq. (1) and comparison with Eq. (8)—leads to ${-}\alpha _{\rm E}^{- 1} = 4{\rm i}{k^{\prime 3}}/3{\epsilon _{\rm h}}$. Full absorption is then possible for a symmetric, dipole-like incident field (coming from both positive and negative $z$’s) interacting with a lossy resonant particle. The required value of ${\rm Im}\{- \alpha _{\rm E}^{- 1}\}$ is then twice the minimum imposed by the optical theorem, therefore entailing an equal contribution of radiative and inelastic losses (the above mentioned critical coupling condition). If we remove the backward component of the incident field (i.e., setting $\beta _{{\textbf{k}_\parallel}\sigma}^ - = 0$) and repeat the above analysis, we find a maximum extinction (i.e., depletion of the incident beam due to both absorption and backscattering) of 50%, provided the incident beam has a dipolar profile [i.e., $\beta _{{\textbf{k}_\parallel}\sigma}^ +$ is shaped as in Eq. (3)], and the particle polarizability also satisfies ${-}\alpha _{\rm E}^{- 1} = 4{\rm i}{k^{\prime 3}}/3{\epsilon _{\rm h}}$.

This analysis can be repeated for a particle responding through a dipolar magnetic polarizability ${\alpha _{\rm M}}$, equally leading to complete absorption and reflection if ${-}\alpha _{\rm M}^{- 1} = 4{\rm i}{k^{\prime 3}}/3{\epsilon _{\rm h}}$ and $2{\rm i}{k^{\prime 3}}/3{\epsilon _{\rm h}}$, respectively, provided the incident light has the magnetic-dipole profile given by Eq. (6). For completeness, we note that the scattered field then has components $\beta _{{\textbf{k}_\parallel}\sigma ,{\rm dip}}^ \pm = (2\pi {\rm i}{\xi _\sigma}{k^2}/{k^\prime _z}\sqrt {{\epsilon _{\rm h}}}) \textbf{m} \cdot {\hat {\textbf{e}}}_{{\textbf{k}_\parallel}\tilde \sigma}^ \pm$, where $\textbf{m}$ is the induced magnetic dipole.

3. PARTICLE NEAR A PLANAR SURFACE

We consider a particle placed at $\textbf{r} = 0$ near a planar surface defined by the $z = a \gt 0$ plane and illuminated with only upward waves of coefficients $\beta _{{\textbf{k}_\parallel}\sigma}^ +$ (i.e., $\beta _{{\textbf{k}_\parallel}\sigma}^ - = 0$, see Fig. 1a). The particle is also affected by field components reflected from the surface with downward coefficients $\beta _{{\textbf{k}_\parallel}\sigma}^ - = ({\beta _{{\textbf{k}_\parallel}\sigma}^ + + \beta _{{\textbf{k}_\parallel}\sigma ,{\rm dip}}^ +}) {\tilde r_{{k_\parallel}\sigma}}$, where $\beta _{{\textbf{k}_\parallel}\sigma ,{\rm dip}}^ +$ is defined in Eq. (9),

A. Maximum Focal Intensity in the Presence of a Planar Surface

The maximal focal field investigated in Section 2.A is modified by the presence of materials (e.g., large field enhancement is produced by coupling to tightly confined plasmons [46]). In the present context, a planar surface contributes to the field with reflected components, so for a beam propagating only along positive $z$ directions (i.e., $\beta _{{\textbf{k}_\parallel}\sigma}^ - = 0$), we can repeat the analysis of Section 2.A with ${\hat {\textbf{e}}}_{{\textbf{k}_\parallel}\sigma}^ +$ replaced by ${\textbf{b}_{{\textbf{k}_\parallel}\sigma}}$, from which we find the condition

B. Maximum Coupling to Surface Polaritons by a Small Particle

In order to investigate maximal coupling to surface modes mediated by the presence of the particle in the configuration of Fig. 1a, we find it convenient to first study the power scattered by the dipole $\textbf{p}$ induced at the particle. The total power emanating from that dipole can be obtained by integrating the radial Poynting vector over a small sphere surrounding it, which yields [43,47] ${{\cal P}^{{\rm scat}}} = 2\omega [{2{{k^\prime}^3}|\textbf{p}{|^2}/3{\epsilon _{\rm h}} + {\rm Im}\{{\textbf{p}^*} \cdot {\textbf{E}^{{\rm ind}}}(0)\}}]$, where ${\textbf{E}^{{\rm ind}}}(0) = {\cal G} \cdot \textbf{p}$ is the electric field induced by the dipole at the particle position. More precisely,

It should be noted that integration over the mode spectral range may also include a background of inelastic absorption, and in addition, it can be problematic when addressing more than one mode. To avoid this, an alternative procedure consists in examining the in-plane far-field associated with each mode (i.e., as emanating from the corresponding pole in such field, expressed as an integral over the surface wave vector) [36,48]. Details of this procedure for a configuration similar to the one here considered are offered in Ref. [36] and can be straightforwardly applied to the present study, although we omit the details because they are not needed in the analysis that follows.

Assuming an optimum incident beam profile as given by Eq. (14), we still have more degrees of freedom to maximize the coupling to the surface: the separation $a$ and the particle polarizability ${\alpha _{\rm E}}$. From Eqs. (12) and (17), we find that, for particles with polarization either parallel ($s = \parallel$) or perpendicular ($s = \bot$) to the surface, ${\alpha _{\rm E}}$ enters ${{\cal P}^{{\rm surf}}}$ through an overall factor $|1/{\alpha _{{\rm E}s}} - {{\cal G}_s}{|^{- 2}}$, and therefore, surface coupling is maximized by minimizing $|1/{\alpha _{{\rm E}s}} - {{\cal G}_s}|$. Additionally, as noted above, the optical theorem imposes [45] ${\rm Im}\{1/{\alpha _{{\rm E}s}}\} \le - 2{k^{\prime 3}}/3 {\epsilon_{\rm h}}$, with the equal sign corresponding to nonabsorbing particles. Also, because ${{\cal P}^{{\rm scat}}}$ cannot be negative, Eq. (16) imposes the inequality ${\rm Im}\{{{\cal G}_s}\} \ge - 2{k^{\prime 3}}/3 {\epsilon_{\rm h}}$. These general constrains of ${\alpha _{\rm E}}$ and ${\cal G}$ imply ${\rm Im}\{1/{\alpha _{{\rm E}s}} - {{\cal G}_s}\} \le 0$, and we conclude that the coupling is maximized for particles that simultaneously satisfy the conditions

In what follows, we consider films that are either optically thick or placed in a symmetric dielectric environment, so all energy coupled to the surface stays trapped in it. We also consider lossless, resonant particles satisfying Eqs. (18a) and (18b), as well as optimized beam profiles given by Eq. (14). For simplicity, we discuss separately particles with their polarizability oriented either parallel ($s =\, \parallel$ and $\textbf{p}\parallel \hat{\boldsymbol z}$) or perpendicular ($s = \bot$ and $\textbf{p}\parallel \hat{\boldsymbol x}$) to the surface. Under these conditions, putting the above expressions together, the maximum fraction of light power coupled to surface modes outside the light cone is (see details in Appendix B)

4. RESULTS AND DISCUSSION

For simplicity, we consider in what follows semi-infinite lossy surfaces and films placed in a symmetric dielectric environment, although we note that part of the power fraction in Eq. (19) could be released as radiation through the far interface of thin films if the permittivity of the dielectric in that region exceeds the one in the near side. Additionally, we evaluate Eqs. (19) and (20) for different types of films and materials (described as explained in Appendix C) and find the optimum distance $a$ at which an absolute value of ${{\cal A}_{{\max}}}$ is achieved as a function of light frequency.

A. Coupling to Plasmons in Drude Metal Films

We study light coupling to plasmons by considering self-standing metallic films described by the Drude permittivity $\epsilon (\omega) = 1 - \omega _{\rm p}^2/\omega (\omega + {\rm i}\gamma)$. The optimum light coupling ${{\cal A}_{{\max}}}$ to plasmons supported by these films is presented as a function of light frequency and particle-surface separation for thick and thin films in Figs. 2a–2d. These plots exhibit sharp features that define an optimum particle-surface separation $a$ for each light frequency $\omega$, which we attribute to the dominant effect of coupling to the metal surface plasmons. Large coupling nearing 100% is observed for both in-plane and out-of-plane orientations of the particle polarizability in thick films, while ${{\cal A}_{{\max}}}$ still takes substantial values for thin films. For reference, the Drude model provides a good description of ultrathin silver and gold films [49] with $\hbar {\omega _{\rm p}} \sim 9$ eV, so for these materials, we have $d = 0.1 c/{\omega _{\rm p}} \approx 2.2\;{\rm nm} $ in Figs. 2c and 2d, a thickness for which plasmons have been experimentally observed in crystalline silver samples [23]. Comparing in-plane and out-of-plane particle polarization, we find that ${{\cal A}_{{\max}}}$ presents a more complex dependence on $a$ and $\omega$ for $\textbf{p}\parallel \hat{\boldsymbol z}$, and in particular, large coupling ${\gt}90\%$ is observed for thick films at $\omega \sim 0.15 {\omega _{\rm p}}$ over a wide range of particle positions. Incidentally, when only using the in-plane polarization of the particle, optimization of both linearly and circularly polarized light fields, which induce dipoles with $\hat{\boldsymbol n} = \hat{\boldsymbol x}$ and $\hat{\boldsymbol n} = (\hat{\boldsymbol x} \pm {\rm i}\hat{\boldsymbol y})/\sqrt 2$ orientations, respectively, gives rise to the exact same maximum coupling.

 

Fig. 2. Optimum coupling to plasmons in Drude metal films. We study the maximum coupling that can be achieved under the configuration of Fig. 1a to surface plasmons in films described through a permittivity $\epsilon (\omega) = 1 - \omega _{\rm p}^2/\omega (\omega + {\rm i}\gamma)$, as calculated for optimized beam profiles depending on frequency, orientation of the particle polarizations, and geometrical parameters. (a–d) Dependence of the maximum coupling fraction ${{\cal A}_{{\max}}}$ on frequency $\omega$ and particle-surface separation $a$ for (a-b) infinite and (c-d) finite film thickness $d$ with the particle polarizability directed along either (a-c) $\hat{\boldsymbol z}$ or (b-d) $\hat{\boldsymbol x}$. (e) Frequency and particle-surface separation for which an absolute maximum coupling is achieved as a function of film thickness. (f) Absolute maximum coupling under the conditions of (e). (g) Drude damping dependence of the absolute maximum coupling for thick and thin films. In (a–f), we take a Drude damping $\gamma = 0.01 {\omega _{\rm p}}$. In (e–g), we show results for both $\hat{\boldsymbol z}$ (red curves) and $\hat{\boldsymbol x}$ (blue curves) particle polarization. In all plots, we consider self-standing films and normalize $a$ to $c/\omega$, $d$ to $c/{\omega _{\rm p}}$, and $\omega$ and $\gamma$ to ${\omega _{\rm p}}$.

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Now, for each film thickness $d$, we search for the values of $a$ and $\omega$ at which ${{\cal A}_{{\max}}}$ reaches an absolute maximum as a function of those parameters. The corresponding optimum values of $a$ and $\omega$ present a strong dependence on $d$ (Fig. 2e), with thicker films producing a larger absolute ${{\cal A}_{{\max}}}$ (Fig. 2f), although this quantity always stays above $50\%$ regardless of how thin the film is (see $\omega$-$a$ maps of ${{\cal A}_{{\max}}}$ for different film thicknesses in Supplement 1, Fig. S2). Incidentally, when moving toward the $d \to 0$ limit, we find that the depletion of material is compensated by incident beam profiles that are strongly weighted near grazing incidence (see Supplement 1, Fig. S3). The metal damping rate $\gamma$ also plays a significant role on the absolute ${{\cal A}_{{\max}}}$ (Fig. 2g), particularly for thin films. Interestingly, $\hat{\boldsymbol x}$ polarization is slightly more efficient in the semi-infinite metal limit with independence of the choice of $\gamma$, while the opposite is true for thin films.

We note that coupling can be improved substantially in asymmetric environments with light incident from the high-index side (see Supplement 1, Fig. S4), reaching nearly 100% for a thin film like that in Figs. 2c and 2d, but now supported on Si ($\epsilon = 12$), which is an experimentally feasible configuration [23].

We also remark that coupling to surface modes is indeed dominated by plasmons, particularly for $\hat{\boldsymbol z}$ dipole orientation, as we show in Supplement 1, Fig. S5 by plotting the integrand of Eqs. (20a) and (20b) under optimum coupling conditions as a function of $q$ and $\omega$, although nonresonant absorption can also play a substantial role in thick metals for $\hat{\boldsymbol x}$ dipole orientation.

B. Coupling to Dielectric Waveguide Modes

High-index planar films provide an attractive approach to guide light because they present comparatively small losses. We consider a film of thickness $d$ and permittivity $\epsilon$ supported on a perfect-electric-conductor (PEC) substrate (Fig. 3a), for which waveguide modes of TE and TM symmetry are confined to the $k \lt {k_\parallel} \lt k\sqrt \epsilon$ region, as shown in Fig. 3b. (We present analogous results for self-standing films in Supplement 1, Fig. S6.) Following the methods discussed in Appendix D, we can efficiently evaluate Eqs. (19) and (20) to yield the coupling maps plotted in Fig. 3c. These plots, which correspond to $\epsilon = 12$ (e.g., Si in the near-infrared spectral region), are universal for any arbitrary film thickness $d$, as this parameter is embedded in the characteristic frequency ${\omega _{{\rm WG}}} = (\pi c/d)/\sqrt {\epsilon - 1}$ (see Appendix D). The dependence of ${{\cal A}_{{\max}}}$ on separation $a$ and frequency $\omega$ is dominated by the dispersion relation of the modes, and in particular, successive modes emerge at periodic intervals determined by ${\omega _{{\rm WG}}}$ as $\omega$ increases. The emergence of every new TE or TM mode produces a sharp feature in the $\omega$-$a$ dependence of the coupling strength for $\hat{\boldsymbol x}$ or $\hat{\boldsymbol z}$ particle polarization, respectively. Although the overall absorption varies slightly with the orientation of the particle dipole, coupling efficiencies ${\gt}90\%$ are observed over broad ranges of the separation $a$ at nearly all frequencies, and complete coupling is also found at frequency-dependent optimum values of $a$. Interestingly, as the number of modes increases with frequency, each of them must receive a smaller fraction of energy from the incident light, which is directly related to the $j$ sums in the expressions for $g_s^{{\rm surf}}$ presented in Appendix D. Conversely, at low frequencies below $0.5 {\omega _{{\rm WG}}}$, there is only one waveguide mode, which has TM symmetry, and for which 100% coupling can be realized when the particle polarizability is oriented along $\hat{\boldsymbol z}$. Further examination of the partial contribution of different modes to the overall coupling (Supplement 1, Fig. S7) reveals that emerging modes of increasingly higher order become dominant as the frequency increases (i.e., the mode that is closest to the light cone for a given frequency generally dominates the coupling of light to the waveguide), and more precisely, for out-of-plane particle dipole orientation, we encounter alternating frequency intervals in which a single mode accounts for nearly 100% coupling. Incidentally, coupling to self-standing Si waveguides is reduced compared to PEC-supported films, but the absolute maximum is still ${\gt}70\%$ (see Supplement 1, Fig. S6).

 

Fig. 3. Complete coupling to dielectric waveguide modes. (a) Scheme of the system under consideration, comprising a planar dielectric waveguide of thickness $d$ and permittivity $\epsilon$, supported on a perfect-electric-conductor (PEC) substrate, and including a neighboring dipolar scatterer at a distance $a$. (b) Dispersion relation of the TE- and TM-polarized waveguide modes supported by this structure, which are confined in the ${k_\parallel}/\sqrt \epsilon \lt \omega /c \lt {k_\parallel}$ region. The axes are normalized using ${\omega _{{\rm WG}}} = (\pi c/d)/\sqrt {\epsilon - 1}$. (c) Optimized light-to-waveguide coupling maps as a function of light frequency and particle-waveguide separation for two different dipole orientations, as indicated by labels. The white contours indicate ${{\cal A}_{{\max}}} = 1$ (100% coupling). Panels (b) and (c) correspond to $\epsilon = 12$ and are universal for any film thickness $d$.

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C. Coupling to Surface Polaritons in 2D Materials: Graphene and hBN

We now move to the optimization of light coupling to 2D polaritons supported by thin films of 2D materials, and specifically, we concentrate on graphene and hBN as prototypical examples of plasmonic and hyperbolic-polaritonic van der Waals materials [20]. For simplicity, we consider self-standing films.

The results that we obtain for graphene (Figs. 4a–4e) are analogous to those for coupling to thin-metal-film plasmons (Figs. 2c and 2d) because the carbon monolayer material can also be well described as a Drude metal with a bulk plasma frequency [50] ${\omega _{\rm p}} = (2e / \hbar)\sqrt {{E_{\rm F}}/d}$, where $d \approx 0.33\;{\rm nm} $ is the atomic layer thickness estimated from the interatomic plane distance in graphite. Remarkably, the optimized photon-to-plasmon coupling efficiency reaches or exceeds $50\%$ over a wide spectral range, in contrast to the much poorer maximum coupling previously predicted when the scattering particle is close to the graphene surface [36]. At low light frequencies, the coupling efficiency reaches even higher values approaching $80\%$ for a particle with $\hat{\boldsymbol z}$ polarization situated at distances of 100’s to 1000’s of nm from the graphene plane (Figs. 4a and 4d). Incidentally, we assume a Drude damping $\hbar \gamma = 2$ meV, so coupling at very low frequencies $\omega \lesssim \gamma$ can be contributed by direct inelastic absorption, rather than plasmon creation, but nevertheless, the features observed in Figs. 4b–4e are dominated by plasmons (see Supplement 1, Fig. S5). Also, an increase in graphene doping generally enhances light coupling to plasmons (cf. Figs. 4b, 4c and Figs. 4d, 4e), as doping plays an effect similar to thickness in metal Drude films (see Fig. 2), bringing the plasmon dispersion relation closer to the light line and reducing the photon-plasmon momentum mismatch.

 

Fig. 4. Optimum coupling to polaritons in 2D materials. (a–e) Coupling to graphene plasmons for two different values of the doping Fermi energy ${E_{\rm F}}$. We consider $\hat{\boldsymbol z}$ and $\hat{\boldsymbol x}$ dipole orientations, as indicated by labels. Plots in (b–e) show the dependence of the coupling fraction ${{\cal A}_{{\max}}}$ on photon energy and dipole-surface separation. Panel (a) shows the maximum coupling for an optimum separation as a function of photon energy. Panels (f–j) are the same as (a–e), but for hBN films of different thickness $d$ (see labels). Vertical dashed lines indicate the limits of the hBN Reststrahlen bands.

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Coupling to hBN films exhibits a rather different behavior because this material only supports phonon polaritons in the two narrow spectral regions known as Reststrahlen bands, which are indicated by vertical dashed lines in Figs. 4f–4j. These bands are distinctly visible in the optimized absorption maps. Outside them, absorption still reaches a maximum of ${\sim}50\%$ under optimized particle and beam-shape conditions as a result of coupling to regular dielectric waveguide modes similar to those of Fig. 3 (i.e., the material behaves as a normal dielectric). Now, although coupling to surface propagating modes only slightly increases absorption above that background level for 1 nm film thickness, we find that thicker hBN films present a strong enhancement inside the Reststrahlen bands, driven by excitation of phonon polaritons and producing an absorption as high as ${\gt}80\%$ for a 100 nm film.

D. Dielectric Particle as a Perfect Scatterer

The above results assume the existence of lossless resonant scatterers that can be placed at optimized distances from the surface. As a practical example of such a type of perfect scatterer, we consider high-index spherical particles, which host strong dipolar Mie resonances of electric and magnetic character that satisfy the conditions of Eqs. (18) to a good approximation (see Supplement 1, Fig. S8). The spectral positions of these resonances depend on the permittivity $\epsilon$ and diameter $D$ of the particle, so these parameters allow us to place the scatterer resonance at the desired frequency for which we intend to optimize absorption. We illustrate this concept by considering a silicon sphere ($\epsilon \approx 12.8$) placed above a semi-infinite silver surface (see Fig. 5a) and choose to orient the induced particle dipole along the $z$ direction using a p-polarized incident beam. We start by computing the separation- and frequency-dependent map of optimized absorption by the metal (Fig. 5b) assuming a lossless resonant point particle. This map presents an absolute maximum at $a \approx 562\;{\rm nm} $ and $\hbar \omega \approx 1.24\; {\rm eV}$ when using an incident beam profile as prescribed by Eq. (14) and graphically plotted in Fig. 1b. We now maintain these parameters fixed (i.e., $a$, $\omega$, and the beam profile), but substitute the point dipole by a finite-size silicon particle and numerically simulate the absorption $A$ and reflection $R$ of this system using a finite-difference frequency-domain method. The resulting dependence of $A$ and $R$ on particle diameter is shown in Fig. 5c in the presence (solid curves) and absence (dashed curves) of the particle. Without the particle, $99\%$ of the incident light is reflected. However, the presence of a silicon sphere produces a resonant absorption feature that reaches $A \gt 80\%$ for a diameter $D \approx 350\;{\rm nm} $. The corresponding electric near-field amplitude distribution further illustrates almost complete reflection in the absence of the particle (Fig. 5d) and strong depletion of reflection when the resonant particle is present (Fig. 5e). We note that, for the resonant particle diameter, a self-standing Si sphere hosts an electric dipole Mie mode at the operating photon energy $\hbar \omega \approx 1.24 \;{\rm eV}$ (see Supplement 1, Fig. S8). These numerical results confirm that high-index dielectric particles constitute good candidates of lossless resonant absorbers for practical implementations in the present context. We note that analogous calculations for different particle diameters show that the maximum absorption increases with decreasing light wavelength (see Supplement 1, Fig. S9).

 

Fig. 5. Optimized absorption for finite-size scatterers. (a) System under consideration, consisting of a silicon sphere ($\epsilon \approx 12.8$) of diameter $D$ placed in front of a semi-infinite silver surface. (b) Optimized absorption as a function of particle-surface separation $a$ and photon energy $\hbar \omega$. (c) Numerically simulated reflection ($R,{R_0}$) and absorption ($A,{A_0}$) for the optimized incident field in the absence (${R_0},{A_0}$) and presence ($R,A$) of the silicon sphere as a function of diameter $D$. Panels (d) and (e) show the out-of-plane electric field amplitude distribution corresponding to the optimized field for the separation and frequency indicated by a white dot in (b) for the parameters indicated by labels in (e). We present numerical simulations in the (d) absence and (e) presence of the particle, with the latter represented by open and solid circles, respectively, the silver surface indicated by horizontal solid lines, and the wavelength shown by the scale bar in panel (d). Incidentally, the plotted out-of-plane field is reduced inside the metal by a factor ${\sim}50$ because of continuity of the normal electric displacement.

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E. Two-Level Atomic Scatterers

In order to evaluate the feasibility of using atomic scatterers to satisfy the conditions imposed by Eqs. (18), we consider a two-level optical resonance of an atom or molecule embedded in a host medium of permittivity $\epsilon$ (e.g., $\epsilon = 1$ for optically trapped atoms in vacuum or $\epsilon \gt 1$ for molecules trapped in a host dielectric under cryogenic conditions that preserve the two-level character of the system [38]). Because the interior of the atom or molecule is not permeated by the host material, assuming that it occupies a small spherical void, the resonance transition dipole needs to be corrected by a depolarization factor ${f_ \epsilon} = 9{\epsilon ^2}/(2 \epsilon + {1)^2}$ [51], which affects the radiative decay rate $\Gamma = {\Gamma _0}{f_ \epsilon}\sqrt \epsilon$ relative to the one in vacuum, ${\Gamma _0}$. In a spectral region near a lossless two-level optical resonance of frequency ${\omega _0}$, we can approximate the atomic polarizability as [52,53]

5. CONCLUSION

In summary, we have demonstrated based on rigorous electromagnetic analytical theory that a small particle placed in front of a polariton-supporting planar surface can boost the coupling of an external focused light beam to the polaritons propagating in the material. Coupling can be complete for waveguide modes in dielectric films, and it also reaches near-unity values for plasmonic materials such as silver and graphene. The conditions for optimum coupling include a suitable particle-surface distance and a shaped light beam profile, for which we offer detailed general prescriptions in closed-form analytical expressions, depending on the light frequency and the surface material and structure. Additionally, the particle must be lossless and resonant, which are feasible conditions met by high-index Mie scatterers and two-level atoms or molecules. In a practical configuration for the near-infrared spectral range, the particle could consist of lithographically carved Si or Ge particles (e.g., spheres or disks), fixed at a designated distance from the surface by an embedding lower-index host medium.

The present results focus on scatterers dominated by an electric dipole resonance, but our analysis can be straightforwardly extended to magnetic dipolar scatterers, for which the beam profile has a different optimum profile, as we illustrate for an isolated particle placed in an infinite homogeneous medium. As an interesting extension of this work, optimum beam profiles could be found for particles having an arbitrary multipolar polarizability, for which we conjecture complete coupling to surface polaritons if they are also made of nonabsorbing materials and exhibit a sufficiently strong optical resonance. While all of these ideas refer to systems operating at a single light frequency, large coupling within an extended frequency range should be possible using those types of particles if they exhibit multiple resonances within that range.

Our study offers a practical route toward the design of localized surface polariton sources at predetermined positions, fed by external shaped light beams and integrated in nanophotonic devices that could be based on planar surfaces and engineered scatterers placed in front of the intended source locations. The same principles apply to two-level atoms, which could be optically trapped at distances ${\sim}100 \;{\rm nm}$ from the surface and serve as mediators of light-polariton coupling as we demonstrate here.